Happy Saint Patrick’s Day! This holiday got me thinking about four-leaf clovers and probability theory. Now, I know that four-leaf clovers are *not* Shamrocks. And, it is shamrocks that are actually associated with St. Patrick’s Day. A shamrock is a young patch of *three-leaf* white clover that grows in winter. Nonetheless, the holiday started me thinking about four-leaf clovers and probabilities.

I’ve heard accounts of people who regularly find four-leaf clovers. They tell stories about finding the rare clovers in minutes. Four-leaf clovers reportedly only occur in 1 out of every 10,000 clovers. I wondered, if four-leaf clovers are rare, how can they find so many? The low probability doesn’t favor them. I don’t disbelieve their stories at all but I wanted to understand how this works.

Fortunately, statistics and probabilities can help! An important aspect of statistics is to ascertain factors that influence the probability of an outcome. In this post, I’ll uncover the relevant factors that help why people can find many four-leaf clovers in spite of the formidable odds.

## Probability and White Clovers

Folklore says that you’ll have good luck if you find a four-leaf white clover (*Trifolium Repens*). White clovers are named after their white flowers. Additionally, the “Trifolium” in their scientific name actually means three-leafed. This is the normal number of leaves for white clover. Other types of clover typically have four leaves, but they don’t give you good luck. So, don’t be fooled by the pretenders!

I’m writing about four-leaf clovers, but I was amazed to discover that white clovers can have five leaves and even more! The Guinness world record is a 56-leaf white clover! The probability of a four-leaf white clover is 1 in 10,000, and the probabilities drop from there. Five-leaf white clovers are 1 in 100,000!

## The Area Necessary to Find a Four-Leaf Clover

The probability of finding a four-leaf clover is 1-in-10,000. This defines the number of times that the event occurs (four-leaf clover) out of a total number of opportunities (all white clovers).

Last summer, I was out in the yard and saw densely packed clovers. Now, I’m sure that these clover invaders in my yard are my neighbor’s fault. It’s more accurate to say that he has bits of grass between his clovers!

At any rate, seeing these large swathes of clovers is what actually prompted me to investigate the four-leaf clover question. How many are likely to be four-leaf clovers? To find one, do you need a large field or will a regular-sized yard suffice? I realized that I had no idea how much area you need to expect to find one four-leaf clover based on the probability.

## Time for Backyard Science!

How much area contains 10,000 clovers? It’s time to do some empirical research—backyard science time with my daughter! This was a great lesson about estimation too.

The first step in our process was to cut out a 6-inch square and lay it on top of a dense patch of clovers. Then, we picked all of the clovers in the patch. We started with the top layer and observed that there were additional layers of clovers underneath. After we handpicked all of the clovers in the square, we counted them.

We counted about 200 clovers in our quarter of a square foot area. To estimate the number of clovers per square foot, we multiplied our count by four (4 X 200 = 800). Then, to calculate the number of square feet necessary for 10,000 clovers, we divided 10,000 by 800. We did the same for the 100,000 clovers you need to expect to find a five-leaf clover.

Drum roll, please! The results are:

- You need 12.5 square feet (1.2 m
^{2}) of a dense clover patch to have 10,000 clovers. - Or, 125 square feet (11.6 m
^{2}) for 100,000 clovers.

The 12.5 square foot area is much smaller than we thought! This is factor number one in explaining how people can find so many four-leaf clovers. One-in-10,000 sounds really intimidating. But, the probability isn’t so bad when a 3’ X 4’ area can contain 10,000 clovers! A small area provides many opportunities to find four-leaf clovers. We’re off to a good start!

## Probability of Finding Four-Leaf Clovers

Probability theory says that the type of event helps determine the probability of the event occurring. We need to figure out whether finding four-leaf clovers are independent or dependent events.

- Events are independent if the probability of the event happening doesn’t change after the event occurs. For example, if you roll a die and get a 6, there is still a 1-in-6 chance of getting a 6 on the next roll.
- Events are dependent if the probability of the event happening does change after the event occurs. For example, if you draw an Ace from a deck of cards and don’t replace it, the chance of drawing an Ace on the next draw decreases.

Determining the type of event in probability theory is really important because it changes our strategy for producing the outcome that we want. I’ve written about dependent events in the context of the Monty Hall problem, where it dramatically changes the strategy for choosing one of three doors.

In the context of finding four-leaf clovers, we need to determine whether finding a four-leaf clover changes the probability of finding another one (dependent events) or does the probability stay the same (independent events). Determining this affects our strategy for how to find them.

## Dependent Probabilities Affect How to Find Four-Leaf Clovers

Scientists at the University of Georgia have identified a gene that permits white clovers to sprout the fourth leaf. Clovers that have this gene still have three leaves typically. But, if the clover experiences certain ecological conditions (pollutants, soil acidity, and temperature), they can produce an extra leaf.

Genes transfer from plant to plant as they reproduce. Therefore, shorter distances between clovers are associated with larger probabilities that they share genes. Likewise, ecological conditions are going to be more similar for plants that are closer together. This all indicates that finding four-leaf clovers are dependent events. When you find one four-leaf clover, you are more likely to find more nearby.

These dependent probabilities explain why websites about finding four-leaf clovers state that four-leaf clovers tend to be in hot spots. If you find one, you might find others pretty quickly! Try revisiting previous hot spots to jump start your search for more.

## How to Find Four-Leaf Clovers

Some websites are dedicated to helping you find four-leaf clovers. Let’s see what they say to determine whether they contain any useful information about finding them quickly.

These websites stress that you don’t want to examine individual clovers. You want to stand and continually scan your eyes over the entire patch. At the same time, use your foot to brush the clovers. This helps ensure that you see the different layers of clovers. This procedure seems to maximize the number of opportunities for observing a rare four-leaf clover in the shortest amount of time. It won’t take long to inspect that 3’ X 4’ area of clover.

In quality control, visual inspectors use a very similar process. The inspectors are very practiced at using the proper scanning method to pick out rare imperfections in a product. When inspectors keep their eyes moving and have fewer “eye fixations”, they tend to find more defects. This is consistent with the four-leaf clover websites!

On the production line, the manufacturer doesn’t just assume that their inspectors will find these rare defects. Instead, they train them with the best practices and develop this as a skill. I think it is a very similar process among those who look for four-leaf clovers. Knowledge and practice help these visual inspectors overcome the low probability of finding rare four-leaf clovers!

## Find Four-Leaf Clovers Quickly and Easily!

Probability theory has helped us understand how to find four-leaf clovers and why some people find them so easily. To recap, here are three important factors that make them easier to find:

- It doesn’t take a large area to observe 10,000 clovers.
- Find a clover hot spot and revisit it frequently.
- Develop and practice your visual inspection skills to find unusual clovers.

Let me know if you find any four-leaf clovers!

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